With Talking Points, I keep finding that the more I push control down into student groups, the better they self-regulate and dive into the material.

So here is a self-guided intro to Talking Points for Algebra 1 students.

Also, with blind students in the classroom, it becomes even more important for equity that student groups speak and listen equitably to ensure inclusion. So with Talking Points, in addition to handouts, I can give a blind student the Word document on a flash drive, they can plug it into a Braille reader (which allows them to read a line at a time), and everybody is off to the races.

The ever-growing Google Drive folder for new sets of math Talking Points is at http://bit.ly/1eHGPWM.

# cheesemonkey wonders

renegotiating the didactic contract

## Friday, July 31, 2015

## Friday, July 24, 2015

### NCTM and The Math Forum Join Forces

Well, here at Twitter Math Camp (#TMC15), this happened today:

Personally, I am thrilled for my good friends at The Math Forum, who have contributed so much to our extensive worldwide professional learning community. But I also want to witness what a milestone it is for TMC that this is the venue at which the merger was announced.

Five years ago, we were a positive but isolated group of individuals connected by Twitter and by our math teaching blogs. Today, our little conference was the platform for an important piece of news in the math education world.

I have said this before — TMC and the MTBoS (the Math Twitter Blog-o-Sphere) are not a flash in the pan. They represent a paradigm shift. We are a movement.

They and The Math Forum are living proof that the "market" does not want what focus groups or policy committees think is the safest generic middle course to follow.

They are proof that what is needed — desperately needed — is a community of individuals committed to embodying a better and more sustainable set of principles in our teaching practice and in our professional development lives:

- Honor the actual work of mathematics teaching that is going on every day — not some sanitized generic ideal that is so removed from reality it cannot be valued.

- Step forward and
that community you wish you could find. As the great psychoanalyst and cantadora Clarissa Pinkola Estès has written, "if you build that community, people will mysteriously show up, announcing that this is exactly what they have been looking for all along."*be*

- Witness and celebrate each other's amazing accomplishments in the classroom, even though the power structure and outside forces refuse to accept the good that we do every day. Cheer each other on. This is about "growing up" as a profession and as a community and accepting that true grown-ups do not wait for permission to do what they know what needs to be done. True grown-ups see what needs to be done and say, "Oh, I see.
do it."*I'll*

- Recognize that this is a movement — and that a movement is what is needed. We have serious problems, but we have phenomenal capacity to respond to what needs to be done. It is easy to stop a few people, but it is impossible to stop a thousand. Remember the motto of #OtterNation:

- Don't take "no" for an answer. As @TrianglemanCSD said in his keynote address today, "Find what you love, and do more of it in your classroom."

## Tuesday, July 14, 2015

### The Primacy-Recency Effect: a conversation with Jennifer Carnes Wilson (episode 1)

Dear Jennifer,

Thanks for engaging with me on David A. Sousa's The Primacy-Recency Effect article. I too like to reread it and think about it around once a year, so your tweet was most timely for me.

There is a Freudian slip-style of typo in his very first sentence that has always struck me as encapsulating the entire debate he has provoked:

On the other hand, if I start right away by framing the big idea, I harness their optimal moment of receptivity and retention, but am I doing so at the risk of their autonomy?

Elizabeth

Thanks for engaging with me on David A. Sousa's The Primacy-Recency Effect article. I too like to reread it and think about it around once a year, so your tweet was most timely for me.

There is a Freudian slip-style of typo in his very first sentence that has always struck me as encapsulating the entire debate he has provoked:

When an individual is processing new information, the amount of information retained depends, among other things onClearly he means to say "when" rather than "what," but for me, that question of "when" versus "what" lies at the very heart of the debate about student discovery of new ideas. Is it more importantit is presented during the learning episode. (emphasis mine)what

*when*students encounter a new idea or*how*they encounter it? If I have students tinker and investigate for too long at the beginning of the class period, I risk missing their window of greatest receptivity and retention.On the other hand, if I start right away by framing the big idea, I harness their optimal moment of receptivity and retention, but am I doing so at the risk of their autonomy?

Elizabeth

## Tuesday, June 30, 2015

### What it means to be a part of a learning community - Tribal Elder Edition

One interesting moment from the Oregon Math Network Conference this past week:

Bill McCallum was leading a large-ish session on building a culture of collaboration through jointly investigating student work.

Fawn (@fawnpnguyen) and I were sitting side by side at a table off to the side, each of us prepping madly for our next sessions. But neither of us could resist the lure of student work. We set our own presentations aside and pulled up the examples of middle school student work on Fawn's computer.

The task for the teachers in the room involved making sense of middle school students' written-out interpretations of different possible takes on how to simplify the expression

7 – 2 ( 3 – 8x)

Being experienced teachers of middle school math students, Fawn and I were both immediately captivated.

"Look at how this student identified right away that the value being distributed is a

I nodded.

I noticed the student's language, which indicated a little mid-process magical thinking about the how to distribute multiplication over subtraction: "...because you use

"The student is using these phrases as magical incantations," I said. "The rules are still spells to him or her." Fawn agreed.

We both recognized these pieces of productive struggle from our own students's journeys. We dissolved into flow as we started talking about different ways to provoke authentic insight and discovery in our students. This is what is fun about getting together with kindred teacher spirits. It gives us the chance to share a deep kind of noticing that happens automatically during the school year, when we are trying to avoid drowning in the sheer overwhelming volume of student work.

While we'd been lost in analyzing, noticing, and wondering — and unnoticed by us — Bill had stepped closer to eavesdrop on our conversation and to join in the fun. At a certain point, he stepped right into the flow of conversation, offering his own noticings and wonderings about the students' wordings and insights. Several times we all burst out laughing — not at the student's work but at our own pure delight in it. Even after all this time, we can all still be captivated by adolescent mathematical thinking.

Well into his late 80s, Michelangelo was often heard to repeat the motto, "Ancora imparo" — "I am still learning." That is a concise summary of the delight that all teachers feel when we get the chance to sit together as a part of a learning community and think about teaching and learning together. This is the best teaching and learning reminder I know, and I always feel blessed when I have one of these flashes of self-remembering during one of these moments. So I wanted to capture this one.

Bill McCallum was leading a large-ish session on building a culture of collaboration through jointly investigating student work.

The task for the teachers in the room involved making sense of middle school students' written-out interpretations of different possible takes on how to simplify the expression

7 – 2 ( 3 – 8x)

Being experienced teachers of middle school math students, Fawn and I were both immediately captivated.

"Look at how this student identified right away that the value being distributed is a

**2 — not just a 2," she said. "They noticed that part of it right away."***negative*I nodded.

I noticed the student's language, which indicated a little mid-process magical thinking about the how to distribute multiplication over subtraction: "...because you use

*order of operations*"; "you always do the problem inside the parentheses first"; "...but then "*it's a problem that*you've got*– 2 on the outside*and – 8x on the inside.""The student is using these phrases as magical incantations," I said. "The rules are still spells to him or her." Fawn agreed.

We both recognized these pieces of productive struggle from our own students's journeys. We dissolved into flow as we started talking about different ways to provoke authentic insight and discovery in our students. This is what is fun about getting together with kindred teacher spirits. It gives us the chance to share a deep kind of noticing that happens automatically during the school year, when we are trying to avoid drowning in the sheer overwhelming volume of student work.

While we'd been lost in analyzing, noticing, and wondering — and unnoticed by us — Bill had stepped closer to eavesdrop on our conversation and to join in the fun. At a certain point, he stepped right into the flow of conversation, offering his own noticings and wonderings about the students' wordings and insights. Several times we all burst out laughing — not at the student's work but at our own pure delight in it. Even after all this time, we can all still be captivated by adolescent mathematical thinking.

Well into his late 80s, Michelangelo was often heard to repeat the motto, "Ancora imparo" — "I am still learning." That is a concise summary of the delight that all teachers feel when we get the chance to sit together as a part of a learning community and think about teaching and learning together. This is the best teaching and learning reminder I know, and I always feel blessed when I have one of these flashes of self-remembering during one of these moments. So I wanted to capture this one.

## Saturday, April 25, 2015

### Impromptu Twitter master class on homework strategies

On Friday afternoon Anna (@borschtwithanna) tweeted out this call for conversation:

```
Hey #MTBoS, teach me how to go over hw. I give answer keys, but it still takes like half the period to answer questions & make connections
```

— Anna Blinstein (@Borschtwithanna) April 24, 2015

What followed was a virtual master class on handling homework in different settings. It's the sort of conversation that the #MTBoS excels at, pulling in thoughtful responses from teachers at every level of practice.

I've been thinking a lot about homework because I am overhauling my homework strategy for next year. This blog post is my attempt to capture my strategy overview, along with pointers to resources that have helped me think through what I need to do.

Overall HW strategy for next year

There are four pillars to my homework strategy for next year:

1. HOMEWORK LAGS CLASSWORK

Henri Picciotto's blog post on this is a classic. There are so many valuable things about having the majority of each night's homework "lag" the current day's classwork. First and foremost, it ensures that most of every night's homework is accessible to every student every day. Secondly, it helps with heterogeneous classes. It gives students multiple at-bats for each kind of homework problem, keeping things meaningful for everybody. New material challenges proficient students, while those who are still working toward proficiency get multiple opportunities to work toward master.

2. DAILY INTRO TASK IS TABLE GROUP HOMEWORK REVIEW

The first ten minutes of class are the time for students to get help on homework — and by "help" I mean helping each other first. A big function of math homework in my view is to help each student cultivate an autonomous and independent approach toward their own struggles with their own problems.

A.H. Almaas describes the problem like this: "Many people... unconsciously act out a desire to be 'saved' by a teacher. But if a teacher 'saved' you, you would lose something. You would lose the value of struggle" (

3. REFRAMING THE ROLE OF THE TEACHER

In keeping with #2, I want to be conscious and intentional with framing my role in their lives about resolving their own problems with problems. Almaas describes this reframing better than anybody I know:

4. COLLECT, STAMP, & GRADE HOMEWORK PACKETS EVERY TWO WEEKS

This has been the most surprisingly successful thing I've done this year. Students have to turn in a stapled packet of their "Home Enjoyment" problems every two weeks. I collect it, I stamp it, and I give them a grade for it every two weeks. Every problem must have been attempted. They should show effort to have sought a resolution for problems they didn't understand the first time through.

This has been a great accountability practice for students. It's an easy, easy 'A' grade every two weeks plus it gives them that push to keep themselves on track and not fall behind on homework. In a high-achieving school filled with motivated students, I expected not to have to do this, but in fact, my experience has revealed the reverse: students appreciate that little push toward accountability. It triggers their automatic reflexes in a way that supports their autonomy.

It also takes me very little time. I am basically just stamping packets, looking for effort and gaps, and rendering a grade whose default is 100% unless stuff is missing or late. I take off a 10% late fee per day. That is usually the only penalty students incur. I've been astounded by how being an old-school hard-ass about this has simplified and streamlined the process.

I hope this is helpful!

I've been thinking a lot about homework because I am overhauling my homework strategy for next year. This blog post is my attempt to capture my strategy overview, along with pointers to resources that have helped me think through what I need to do.

Overall HW strategy for next year

There are four pillars to my homework strategy for next year:

- HOMEWORK LAGS CLASSWORK: I am implementing Henri Picciotto's strategy of having the majority of each night's homework "lag" the current classwork focus of the day;
- EACH DAY'S INTRO TASK IS TABLE GROUP REVIEW: the first ten minutes of class will be for students to discuss/review/help each other out on the previous night's homework problems together in their table groups;
- REFRAMING THE ROLE OF THE TEACHER: I will take questions on particularly troublesome homework problems, but I will take
*whole-group questions only*; and - COLLECT, STAMP, & GRADE HW PACKETS EVERY 2 WEEKS: I collect, stamp, and grade homework packets every two weeks, grading for completeness of effort (every problem in every night's problem set attempted).

1. HOMEWORK LAGS CLASSWORK

Henri Picciotto's blog post on this is a classic. There are so many valuable things about having the majority of each night's homework "lag" the current day's classwork. First and foremost, it ensures that most of every night's homework is accessible to every student every day. Secondly, it helps with heterogeneous classes. It gives students multiple at-bats for each kind of homework problem, keeping things meaningful for everybody. New material challenges proficient students, while those who are still working toward proficiency get multiple opportunities to work toward master.

2. DAILY INTRO TASK IS TABLE GROUP HOMEWORK REVIEW

The first ten minutes of class are the time for students to get help on homework — and by "help" I mean helping each other first. A big function of math homework in my view is to help each student cultivate an autonomous and independent approach toward their own struggles with their own problems.

A.H. Almaas describes the problem like this: "Many people... unconsciously act out a desire to be 'saved' by a teacher. But if a teacher 'saved' you, you would lose something. You would lose the value of struggle" (

*Diamond Heart Book One,*p. 123). In my view, the first ten minutes of class are the place where I expect students to begin to shift their mindset about the value of struggle. If a problem is not pushing beyond their Zone of Proximal Development, I expect them to develop the habit of resolving their own confusions themselves.3. REFRAMING THE ROLE OF THE TEACHER

In keeping with #2, I want to be conscious and intentional with framing my role in their lives about resolving their own problems with problems. Almaas describes this reframing better than anybody I know:

So there are two ways to approach the teacher. One approach is to hope the teacher will take away your problems; the other is to use the teacher, not with the expectation that she will take away your problems or offer solutions or "make it better" but that she will give you a little push in your struggle. (This is why I love Dan Anderson's (@dandersod) description of the "mass confusion" rule: unless a problem causes mass confusion, students have to work out their problems independently and help each other out on during that first ten minutes of class.DH Book One,p. 124).

4. COLLECT, STAMP, & GRADE HOMEWORK PACKETS EVERY TWO WEEKS

This has been the most surprisingly successful thing I've done this year. Students have to turn in a stapled packet of their "Home Enjoyment" problems every two weeks. I collect it, I stamp it, and I give them a grade for it every two weeks. Every problem must have been attempted. They should show effort to have sought a resolution for problems they didn't understand the first time through.

This has been a great accountability practice for students. It's an easy, easy 'A' grade every two weeks plus it gives them that push to keep themselves on track and not fall behind on homework. In a high-achieving school filled with motivated students, I expected not to have to do this, but in fact, my experience has revealed the reverse: students appreciate that little push toward accountability. It triggers their automatic reflexes in a way that supports their autonomy.

It also takes me very little time. I am basically just stamping packets, looking for effort and gaps, and rendering a grade whose default is 100% unless stuff is missing or late. I take off a 10% late fee per day. That is usually the only penalty students incur. I've been astounded by how being an old-school hard-ass about this has simplified and streamlined the process.

I hope this is helpful!

## Sunday, April 19, 2015

### The deeper wisdom of the body in math class

*This post is for Malke.*

As I was eavesdropping on a recent conversation between Lani Horn (@tchmathculture) and Malke Rosenfeld (@mathinyourfeet), I received a pointer to an article summarizing recent research that shows that kids with ADHD actually need to squirm in order to learn.

This makes sense to me. The deeper wisdom of the body is usually overlooked in thinking about teaching, learning, and assessment in mathematics. And yet, it can provide a vital link for our students in claiming their mathematics as well as their humanity.

I was thinking about this on Friday afternoon when I tweeted out the following:

I love the dulcet tones of compasses, rulers, & pencils during a Friday afternoon constructions quiz. #Malke tweeted back:geomchat

I love that they are doing all that by hand. And that there are dulcet tones. :)I responded:

Geo has affirmed my belief in the life made by hand. Huge benefits to Ss [students] from physically constructing their understanding.

I had another in a series of lightbulb moments this past month about what How People Learn says about externalizing our understanding.And ever the good online learning partner, Malke tweeted back:

have you blogged about it?So here I am.

In

*How People Learn*, the authors talk about how we can use skits, presentations, and posters in group work to help students externalize their emerging understanding. This makes sense to me. In order to learn something, one first has to notice it, and that means developing a metacognitive self-awareness of the process and how it’s going.

Over the last few years, I have found that teaching students to use foldables, INBs (Interactive Notebooks), guided note-taking, and physical constructions is another extremely rich field of helping students to externalize their emerging understanding — only in these cases, they are externalizing their understanding through physical, kinesthetic processes — not just through talk, listening, and presentation processes.

The physical dimension is a good grounding for conceptual understanding. Teaching students how to literally use their tools can be a multidimensional process of making their learning both physical and tangible. Flipping open a flap or a page in a composition book is a physical manifestation of the process of retrieval or comparison or evaluation. Likewise, the process of using patty paper as a tracing medium to externalize the concept of superposition and projection of a figure to confirm congruence is a way of helping students to slow down their speeding monkey minds and to become present with the mathematics that are right in front of them.

When I tuned in to the clatter of compasses, rulers, and pencils on Friday, I really noticed how deeply engaged my students were with the geometry they were working on. Their body postures indicated how deeply immersed they were in the experience of flow: set your compass opening to an appropriate width and draw an arc across the angle you want to copy. Stab the endpoint of the segment where you want to create a new copy of your original angle and swipe the same arc there. Go back to the first angle. Refine your compass opening so that it now matches the width between the intersections of your arc and the original angle. Shift your paper, stab at the lower intersection of your copied angle-in-process and swipe an arc that will intersect with the arc you just drew there. Drop your compass; pick up your straight edge. Carefully draw a line to connect the endpoint of your target segment with the intersection of the arcs you have drawn, completing the terminal side/ ray you need to draw. Drop the straight edge; position the patty paper over your original angle and use your straight edge to trace it. Drop the straight edge and carefully slide your traced angle over your constructed angle. Does it match your figure perfectly?

The concentration etched on their brows matched the precision of their work on the page in front of them. Bisect an angle. Construct the perpendicular bisector of a segment. Construct a parallel line through an external point by using it to define an angle that you can copy.

Hopping from one stone to the next, you can cross an entire river. By placing one foot on the Earth after another in a pattern of glide reflections, you can complete a journey of a thousand miles or more.

That is one of the lessons of deductive and spatial reasoning at the heart of any good Geometry course. Noticing that it is happening in my classroom — really happening through physical, mental, and whole-hearted engagement — is one of the greatest blessings of being a teacher.

## Tuesday, April 14, 2015

### 40 days and nights of standardized testing: a reflection

And on the 289th day, God made brownie mix. He divided the mix into the batter & the topping. And God saw the brownie mix, that it was good. And God said, Let the brownie mix be cheap and abundant and distributed to all the grocery stores so that all those who suffer, may make brownies. And the baked brownies He called "one serving," and he forbade the posting of the nutrition facts. And there were brownies, later with toppings, a comfort food.

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